Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture

Abstract

This study is based on the trivial transcription of the vertices of a Boolean textit{N}-Dimensional Hypercube $textbf{H}_{N} $ into a subset $mathbb{S}_{N}$ of the decimal natural numbers $mathbb{N}.$ Such straightforward mathematical manipulation permits to achieve a recursive construction of the whole set $mathbb{N}.$ In this proposed scheme, the Mersenne numbers act as upper bounds of the iterative building of $mathbb{S}_{N}$. The paper begins with a general description of the Collatz or $left(3x+1right)$ algorithm presented in the $mathbb{S}_{N} subset mathbb{N}$ iterative environment. Application of a defined textit{ad hoc} Collatz operator to the Boolean Hypercube recursive partition of $mathbb{N}$, permits to find some hints of the behavior of natural numbers under the $left(3x+1right)$ algorithm, and finally to provide a scheme of the Collatz conjecture partial resolution by induction.